(i) All questions are compulsory.
(ii) Questions 1 to 7 are very short answer type questions. These questions carry one mark each.
(iii) Questions 8 to 10 are short answer type questions. These questions carry four marks each.
(iv) Question 11 is long answer type question. This question carries six marks.
Question 1 ( 1.0 marks)
Letf: R→ Rbe a function defined by f(x) = 2x+ 3. Find a function g: R → R satisfying the condition gof= fog= IR.
Question 2 ( 1.0 marks)
A function
f:
Z→
A, where
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u4g-CBZez2QmYvrkJXaa7mHKUvhzgjuJp4QEzhdjF9vkyLdoeQXtPVdoqhlAeMNW79GCYY8wkx6shsyi_47n6hFhtKW197xT92i2VAeq58DSziqJ-G6vrJP8PshF8oba1xw6AlPfJZmQi6F639jYKTCw=s0-d)
is defined as
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uBRufbq5t1htsdWPhr6yd4FolBANH7zcDTDdSIMJ0y9r7pr2mM_SGUjG6m0HVU-82XxCpZcBDW5w0y9of7xme8xv-2p2JOWh0kq-VdXpAaJEKHXpeFYmf7exOVORY7Xl6c8pu9FDI9GbxaIlWa-zSV=s0-d)
.
Without finding the actual inverse of
f, show that
fis bijective.
Question 3 ( 1.0 marks)
Let
X= {1, 2, 3, 5, 6, 10, 15, 30}
Operations ‘*’, ‘
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uAgrEnqHgx9V9d1U2xM5C-iz-e4eilds0iYvscNx6gn4cl_VK9I2AFTTgmmfh30HaXhv2eEsqRS_zYl1TF1J2l0Q3IpJ3rHwhF-PZpZaXXfuc3zHITknDH-c-uEVeRwIbLw330N9xgk7ESIjeIPF8rBDE=s0-d)
’ and ‘Δ’ on
Xare defined as follows:
a*
b= LCM (
a, b)
a
b= HCF (
a, b)
aΔ
b=
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_utwsM0MyxwCsXgQRroqzC9p6YdY2lxrE8r_K7p5ptAuXrePPIrED0Xkdtbi5JbS25OjpPK2uomVyi8nuv659s6vV48x722clJ9EdzF8qk80-fP8UeYwldqvbjMwlGPDdfUiaaOrr5k72YqQPvMv_w_J0g=s0-d)
Which of the given operations is (are) binary operation(s)?
Question 4 ( 1.0 marks)
If
f:N→
Nis one-one and onto function, then find the value of the following expression.
Question 5 ( 1.0 marks)
Letf be any non-zero real valued function and let gbe a function given by g(x) = (k+ 2)x. Find the value of ksuch that gof = 2f.
Question 6 ( 1.0 marks)
If Y= {x: x∈ Nand x≤ 4}, then show that the relation Ron Ydefined by R= {(a − b): |a − b| is a multiple of 3} is nota trivial relation.
Question 7 ( 1.0 marks)
A function f: R→ Ris defined as f(x) = 3x−5.
Find αsuch that f(α) = f−1(α).
Question 8 ( 4.0 marks)
A function
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vRpwYr69xIWDyvk6kopxFloVZgPZN6YyCDhVrdMTxD84JDbH8Oh9h_8D7W6CQ4ugkaH-PAwoNbnYaERjRg83OzRY-ua73C22hTDcwlrkV9a00xd-5259gUMy6Hm3infD8umL5eQ4bMIf2PTm0r_tIEiA=s0-d)
is defined as
f(
x) =
x2− 3
x+ 2.
- By not calculating the actual inverse of f show that f is invertible.
- Find the inverse of f.
Question 9 ( 4.0 marks)
Let
A=
R− {0} and
B=
A ×
R.
Let the binary operation * on
Bbe defined as (
a, b) * (
c, d) =
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vTHPlTzMJsR1TrZ31mXejAlvMDOvB34SLb86zfCGmAOggNL8V5I0RxYyqN3WCsmbI3rwKogDiRExS8jeT9bcetZiLQiX7wjrwlKarYqh0I7tXXeRInYPvpaiDKmFR6Vz9xSgAq8S8A5pminjgPPOSRoA=s0-d)
.
(i) Find the identity element of
Bwith respect to the operation *.
(ii) Show that set
Bis invertible with respect to the operation *.
Also, find the inverse of (−5, 3).
Question 10 ( 4.0 marks)
Let
Q+denote the set of all positive rational numbers.
A relation
Ron
Q+×
Q+is defined as
![](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sgYiUTU4yI6NTO9HI6sWzayhPw9zTBraBLseBBrPR0HUizp31D34gHA0XYNUU7mnD7cyVJKkPAw6FNi-3cOTW4_UCIsT6bsrDfSqplIcCecCT-awIegHroH-ItAtI9GCfxwquUWPmR34r9FwYqIUCcdVs=s0-d)
.
Show that
Ris an equivalence relation.
Question 11 ( 6.0 marks)
Two real valued functions fand gare defined as f(x) = e2xand g(x) = x− 3. Show that the functions
are one-to-one and onto, and thus, find their inverse.